Transcript CBE 150A – Transport Spring Semester 2014 Flow Around Objects

Flow Around Immersed Objects
Incompressible Flow
CBE 150A – Transport
Spring Semester 2014
Goals
• Describe forces that act
on a particle in a fluid.
• Define and quantify the
drag coefficient for
spherical and nonspherical objects in a flow
field.
• Define Stokes’ and
Newton’s Laws for flow
around spheres.
CBE 150A – Transport
Spring Semester 2014
Flow Around Objects
There are many processes that involve flow through a
porous medium such as a suspension of particles:
CBE 150A – Transport
Spring Semester 2014
Flow Around Objects
There are many processes that involve flow around
objects - some are more interesting than others:
CBE 150A – Transport
Spring Semester 2014
Forces
Dynamic
Fk results from the relative motion
of the object and the fluid (shear
stress)
Static
Fs results from external pressure
gradient (Fp) and gravity (Fg).
M 2  M 1   F  Fk  Fg  Fp
CBE 150A – Transport
Spring Semester 2014
Dynamic Forces
For flow around a submerged object a drag coefficient
Cd is defined:
Cd A u
Fk 
2
2
0
U0 is the approach velocity (far from the object), ρ is
the density of the fluid, A is the projected area of the
particle, and Cd is the drag coefficient analogous to
the friction factor in pipe flow (keep this in mind).
CBE 150A – Transport
Spring Semester 2014
Projected Area
The projected area used in the Fk is the area “seen” by
the fluid.
Spherical Particle
A  R 
2
CBE 150A – Transport
D
2
4
Spring Semester 2014
Projected Area
For objects having shapes other than spherical, it is
necessary to specify the size, geometry and orientation
relative to the direction of flow.
Cylinder
Axis perpendicular to flow
Axis parallel to flow
CBE 150A – Transport
Rectangle
Circle
A  LD
A
D
2
4
Spring Semester 2014
Drag Coefficient
The drag coefficient, like the friction factor in pipes
depends on the Reynolds number
Re 
Du 0 

D is particle diameter or a characteristic length and ρ
and μ are fluid properties.
CBE 150A – Transport
Spring Semester 2014
Drag Coefficient
For slow flow around a sphere and Re<10
24
24 
Cd 

Re Du 0 
Recall:
Cd A u
Fk 
2
2
0
Stokes’ Law for Creeping Flow Around Sphere
Fk  3 Du 0
CBE 150A – Transport
Spring Semester 2014
Drag Coefficient
Re  10 Cd  24 Re
CBE 150A – Transport
Re  1000 Cd  0.44
Spring Semester 2014
Why Different Regions?
As the flow rate increases wake drag becomes an important factor. The
streamline pattern becomes mixed at the rear of the particle thus causing
a greater pressure at the front of the particle and thus an extra force term
due to pressure difference. At very high Reynolds numbers completely
separate in the wake.
Streamline separation
CBE 150A – Transport
Spring Semester 2014
Drag Coefficient Adjustment
Russian Shkval Torpedo
CBE 150A – Transport
Spring Semester 2014
Determining Flow Fields
CBE 150A – Transport
Spring Semester 2014
Videos
CBE 150A – Transport
Spring Semester 2014
Static Forces
Static forces exist in the absence of fluid motion. They
include the downward force of gravity and the upward
force of buoyancy that results from the gravity induced
pressure gradient in the z-direction
 Fg
P1
Fg  m p g  V p  p g
Fb  AP2  P1   A f gh 
P2  P1   f gh
 Vp  f g
 Fb
CBE 150A – Transport
Spring Semester 2014
Total Force
The gravity and buoyancy forces on an object immersed
in liquids do not generally balance each other and the
object will be in motion.
 Fg
Ft  Fk  Fg  Fb
What is the direction of Fk?
 Fb
CBE 150A – Transport
It is always opposite to the direction of
particle motion
Spring Semester 2014
Equilibrium
When a particle whose density is greater than that of the
fluid begins to fall in response to the force imbalance, it
begins to accelerate (F=ma). As the velocity increases
the viscous drag force also increases until all forces are in
balance. At this point the particle reaches terminal
velocity.
0  Fk  Fg  Fb
2
t
u
 F 0  Cd Ap  f 2  m p g  Vp  f g
CBE 150A – Transport
Spring Semester 2014
Terminal Velocity
General Expression: If the particle has a uniform
density, the particle mass is Vpp and
u
0  Cd Ap  f
 V p  f   g g
2
2
t
ut 
4 g  p   f  D p
3Cd  f
Use: Falling ball viscometer to measure viscosity
CBE 150A – Transport
Spring Semester 2014
Settling Velocity
Stokes’ Region: The settling (terminal) velocity of small particles
is often low enough that the Reynolds number is less than unity
(Cd = 24/Re).
ut 
D  p   f g
2
p
18
Re  1
Newton’s Region: Between 1000<Re<200,000 Cd = 0.44
ut  1.75
gD p  p   f 
f
Fk  0.055 Dp2ut2  f
Note: Intermediate flow requires iteration
CBE 150A – Transport
Spring Semester 2014
Criterion for Settling Regime
Reynolds number is a poor criteria for determining the
proper regime for settling. We can derive a value K that
depends solely on the physical parameters
 g f  p   f 
K  Dp 

2



13
K < 2.6
Stokes’ Law
K > 68.9
Newton’s Law
CBE 150A – Transport
Spring Semester 2014
Example
A cylindrical bridge pier 1 meter in diameter is submerged
to a depth of 10m in a river at 20°C. Water is flowing past
at a velocity of 1.2 m/s. Calculate the force in Newtons on
the pier.
 water  998.2kg m3
 water  1.005  x103 kg m  s
CBE 150A – Transport
u0  1.2m s
Spring Semester 2014
Example
Cd A u02
Fk 
2
 u0 D 998.2 kg m3 1.2 m s 1m
6
Re 


1
.
192

10

1.005 103 kg m  s
Fig. 7.3 gives Cd ≈ 0.35
Projected Area = DL = 10 m2
2
0.35
kg
m
2
Fk 
10m 2  998.2 3  1.2 2  2,515 N
2
m
s
CBE 150A – Transport
Spring Semester 2014
Example
Estimate the terminal velocity of limestone particles (Dp = 0.15 mm,  = 2800
kg/m3) in water @ 20°C.
CBE 150A – Transport
Spring Semester 2014
Example
Guess Re = 4
ut 
4  9.8
Cd = 16.0 – from Figure 7.6
m
kg
4



2800

998
.
2

1
.
5

10
m
2
3
m
s
m
 0.015
kg
s
3 16.0  998.2 3
m
1.5 10 4 m  998.2 kg m3  0.015 m s
Re 
 2.2
3
1.005 10 kg m  s
CBE 150A – Transport
Spring Semester 2014
Example
Guess Re = 2
Cd = 22
Ut = 0.013 m/s
Re = 1.9
CBE 150A – Transport
Spring Semester 2014
10 Minute Problem
Tiger Woods is practicing putting golf balls on a cruise ship, he
makes a slight miscalculation and the ball rolls off the “green” and
falls into the ocean. Assuming the ball quickly attains its terminal
velocity and the descent is defined by the Newton’s law flow
regime, how long does it take the ball to hit the ocean floor 300 m
below ?
Golf ball data:
Diameter = 43 mm
Weight = 45 grams
Density = 1.16 g /cm3
Seawater data: Density = 1.025 g /cm3
Viscosity = 0.01 g / cm sec
CBE 150A – Transport
Spring Semester 2014